= Solution
The array has $K=10$ replicates, $I=2$ levels of factor $a$, and $J=3$ levels of factor $b$, giving $n=60$. R vectorizes an array with the first index changing fastest, then the second, then the third. Accordingly, `gl(I,K,length(y))` repeats each $a$ level for the ten replicates and cycles over its two levels, while `gl(J,K*I,length(y))` repeats each $b$ level for twenty observations. These factors align with the vectorized response.
The first fit is the additive <Analysis of variance> model $Y_{kij}=\mu+\alpha_i+\beta_j+\varepsilon_{kij}$, with independent normal errors of common <variance> $\sigma^2$. Under the default treatment constraints $\alpha_1=\beta_1=0$, the four fitted coefficients estimate the baseline cell mean, the $a=2$ minus $a=1$ effect, and the $b=2,3$ contrasts with $b=1$. Thus the baseline fitted mean is $3.7673$, the second $a$ level adds $3.4542$, and the second and third $b$ levels add $-6.3215$ and $-5.8268$.
The summary $t$ tests compare each indicated coefficient with zero, using its <standard error> and residual <variance> estimate $77.20/56$. They have 56 residual <degrees of freedom>. The three non-intercept contrasts are strongly significant; the intercept test asks whether the baseline mean is zero, not whether either factor matters overall. The <Analysis of variance> tests are joint nested-model $F$ tests: no $a$ main effect, with one numerator degree of freedom, and no $b$ main effect, with two. Their sequential sums of squares are $178.98$ and $494.39$. In this balanced design the main-effect subspaces are orthogonal, so ordering $a,b$ does not change those sums of squares. The denominator is the same residual mean square $77.20/56$; both factor effects are highly significant.
The second fit expands `a*b` to `a+b+a:b`, adding two <interaction> coefficients. It allows the $a$ effect to differ between $b$ levels and has six parameters, equivalently one mean per cell, with 54 residual <degrees of freedom>. The <interaction> adds only about $0.27$ to explained sum of squares; its $F$ statistic is $0.0963$ with <degrees of freedom> $(2,54)$ and $p=0.9084$. There is no evidence against additivity, so the simpler first model is preferable on these data. This does not prove the true <interaction> is exactly zero.
For the final expression, $p=6$, $p_0=4$, and $n=60$. The fitted vectors are orthogonal projections onto nested model spaces, so $\|P Y-P_0Y\|^2$ is the additional sum of squares due to <interaction>. The final code computes
$$
\boxed{F=\frac{\|PY-P_0Y\|^2/2}{\|y-PY\|^2/54}=0.0963\ \text{approximately},}
$$
the same <interaction> test as the <Analysis of variance> row. The exact value uses the original, unrounded observations and fits. Using only the displayed rounded sums $0.27$ and $76.93$ gives about $0.0948$ instead; that discrepancy is rounding, not a different test. The omitted observations prevent reconstructing more digits from the printed data alone.
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